On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(mathbb {H}^n), min mathbb {N}^n,$ under the heat kernel transform on $mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(mathbb {H}^n), s(>0) in mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $mathbb {H}^n$ under the heat kernel transform.